Step | Hyp | Ref
| Expression |
1 | | elex 3185 |
. . 3
⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V) |
2 | 1 | anim2i 591 |
. 2
⊢ ((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) → (𝐺 ∈ Grp ∧ 𝑆 ∈ V)) |
3 | | gaid.1 |
. . . . . . . 8
⊢ 𝑋 = (Base‘𝐺) |
4 | | eqid 2610 |
. . . . . . . 8
⊢
(0g‘𝐺) = (0g‘𝐺) |
5 | 3, 4 | grpidcl 17273 |
. . . . . . 7
⊢ (𝐺 ∈ Grp →
(0g‘𝐺)
∈ 𝑋) |
6 | 5 | adantr 480 |
. . . . . 6
⊢ ((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) → (0g‘𝐺) ∈ 𝑋) |
7 | | ovres 6698 |
. . . . . . 7
⊢
(((0g‘𝐺) ∈ 𝑋 ∧ 𝑥 ∈ 𝑆) → ((0g‘𝐺)(2nd ↾ (𝑋 × 𝑆))𝑥) = ((0g‘𝐺)2nd 𝑥)) |
8 | | df-ov 6552 |
. . . . . . . 8
⊢
((0g‘𝐺)2nd 𝑥) = (2nd
‘〈(0g‘𝐺), 𝑥〉) |
9 | | fvex 6113 |
. . . . . . . . 9
⊢
(0g‘𝐺) ∈ V |
10 | | vex 3176 |
. . . . . . . . 9
⊢ 𝑥 ∈ V |
11 | 9, 10 | op2nd 7068 |
. . . . . . . 8
⊢
(2nd ‘〈(0g‘𝐺), 𝑥〉) = 𝑥 |
12 | 8, 11 | eqtri 2632 |
. . . . . . 7
⊢
((0g‘𝐺)2nd 𝑥) = 𝑥 |
13 | 7, 12 | syl6eq 2660 |
. . . . . 6
⊢
(((0g‘𝐺) ∈ 𝑋 ∧ 𝑥 ∈ 𝑆) → ((0g‘𝐺)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥) |
14 | 6, 13 | sylan 487 |
. . . . 5
⊢ (((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) → ((0g‘𝐺)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥) |
15 | | simprl 790 |
. . . . . . . 8
⊢ ((((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑦 ∈ 𝑋) |
16 | | simplr 788 |
. . . . . . . 8
⊢ ((((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑥 ∈ 𝑆) |
17 | | ovres 6698 |
. . . . . . . . 9
⊢ ((𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑆) → (𝑦(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑦2nd 𝑥)) |
18 | | df-ov 6552 |
. . . . . . . . . 10
⊢ (𝑦2nd 𝑥) = (2nd
‘〈𝑦, 𝑥〉) |
19 | | vex 3176 |
. . . . . . . . . . 11
⊢ 𝑦 ∈ V |
20 | 19, 10 | op2nd 7068 |
. . . . . . . . . 10
⊢
(2nd ‘〈𝑦, 𝑥〉) = 𝑥 |
21 | 18, 20 | eqtri 2632 |
. . . . . . . . 9
⊢ (𝑦2nd 𝑥) = 𝑥 |
22 | 17, 21 | syl6eq 2660 |
. . . . . . . 8
⊢ ((𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑆) → (𝑦(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥) |
23 | 15, 16, 22 | syl2anc 691 |
. . . . . . 7
⊢ ((((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥) |
24 | | simprr 792 |
. . . . . . . . 9
⊢ ((((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑧 ∈ 𝑋) |
25 | | ovres 6698 |
. . . . . . . . . 10
⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑆) → (𝑧(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑧2nd 𝑥)) |
26 | | df-ov 6552 |
. . . . . . . . . . 11
⊢ (𝑧2nd 𝑥) = (2nd
‘〈𝑧, 𝑥〉) |
27 | | vex 3176 |
. . . . . . . . . . . 12
⊢ 𝑧 ∈ V |
28 | 27, 10 | op2nd 7068 |
. . . . . . . . . . 11
⊢
(2nd ‘〈𝑧, 𝑥〉) = 𝑥 |
29 | 26, 28 | eqtri 2632 |
. . . . . . . . . 10
⊢ (𝑧2nd 𝑥) = 𝑥 |
30 | 25, 29 | syl6eq 2660 |
. . . . . . . . 9
⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑆) → (𝑧(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥) |
31 | 24, 16, 30 | syl2anc 691 |
. . . . . . . 8
⊢ ((((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑧(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥) |
32 | 31 | oveq2d 6565 |
. . . . . . 7
⊢ ((((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦(2nd ↾ (𝑋 × 𝑆))(𝑧(2nd ↾ (𝑋 × 𝑆))𝑥)) = (𝑦(2nd ↾ (𝑋 × 𝑆))𝑥)) |
33 | | simpll 786 |
. . . . . . . . 9
⊢ (((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) → 𝐺 ∈ Grp) |
34 | | eqid 2610 |
. . . . . . . . . . 11
⊢
(+g‘𝐺) = (+g‘𝐺) |
35 | 3, 34 | grpcl 17253 |
. . . . . . . . . 10
⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑦(+g‘𝐺)𝑧) ∈ 𝑋) |
36 | 35 | 3expb 1258 |
. . . . . . . . 9
⊢ ((𝐺 ∈ Grp ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦(+g‘𝐺)𝑧) ∈ 𝑋) |
37 | 33, 36 | sylan 487 |
. . . . . . . 8
⊢ ((((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦(+g‘𝐺)𝑧) ∈ 𝑋) |
38 | | ovres 6698 |
. . . . . . . . 9
⊢ (((𝑦(+g‘𝐺)𝑧) ∈ 𝑋 ∧ 𝑥 ∈ 𝑆) → ((𝑦(+g‘𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = ((𝑦(+g‘𝐺)𝑧)2nd 𝑥)) |
39 | | df-ov 6552 |
. . . . . . . . . 10
⊢ ((𝑦(+g‘𝐺)𝑧)2nd 𝑥) = (2nd ‘〈(𝑦(+g‘𝐺)𝑧), 𝑥〉) |
40 | | ovex 6577 |
. . . . . . . . . . 11
⊢ (𝑦(+g‘𝐺)𝑧) ∈ V |
41 | 40, 10 | op2nd 7068 |
. . . . . . . . . 10
⊢
(2nd ‘〈(𝑦(+g‘𝐺)𝑧), 𝑥〉) = 𝑥 |
42 | 39, 41 | eqtri 2632 |
. . . . . . . . 9
⊢ ((𝑦(+g‘𝐺)𝑧)2nd 𝑥) = 𝑥 |
43 | 38, 42 | syl6eq 2660 |
. . . . . . . 8
⊢ (((𝑦(+g‘𝐺)𝑧) ∈ 𝑋 ∧ 𝑥 ∈ 𝑆) → ((𝑦(+g‘𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥) |
44 | 37, 16, 43 | syl2anc 691 |
. . . . . . 7
⊢ ((((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑦(+g‘𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥) |
45 | 23, 32, 44 | 3eqtr4rd 2655 |
. . . . . 6
⊢ ((((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑦(+g‘𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑦(2nd ↾ (𝑋 × 𝑆))(𝑧(2nd ↾ (𝑋 × 𝑆))𝑥))) |
46 | 45 | ralrimivva 2954 |
. . . . 5
⊢ (((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) → ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦(+g‘𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑦(2nd ↾ (𝑋 × 𝑆))(𝑧(2nd ↾ (𝑋 × 𝑆))𝑥))) |
47 | 14, 46 | jca 553 |
. . . 4
⊢ (((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) → (((0g‘𝐺)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦(+g‘𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑦(2nd ↾ (𝑋 × 𝑆))(𝑧(2nd ↾ (𝑋 × 𝑆))𝑥)))) |
48 | 47 | ralrimiva 2949 |
. . 3
⊢ ((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) → ∀𝑥 ∈ 𝑆 (((0g‘𝐺)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦(+g‘𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑦(2nd ↾ (𝑋 × 𝑆))(𝑧(2nd ↾ (𝑋 × 𝑆))𝑥)))) |
49 | | f2ndres 7082 |
. . 3
⊢
(2nd ↾ (𝑋 × 𝑆)):(𝑋 × 𝑆)⟶𝑆 |
50 | 48, 49 | jctil 558 |
. 2
⊢ ((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) → ((2nd ↾ (𝑋 × 𝑆)):(𝑋 × 𝑆)⟶𝑆 ∧ ∀𝑥 ∈ 𝑆 (((0g‘𝐺)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦(+g‘𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑦(2nd ↾ (𝑋 × 𝑆))(𝑧(2nd ↾ (𝑋 × 𝑆))𝑥))))) |
51 | 3, 34, 4 | isga 17547 |
. 2
⊢
((2nd ↾ (𝑋 × 𝑆)) ∈ (𝐺 GrpAct 𝑆) ↔ ((𝐺 ∈ Grp ∧ 𝑆 ∈ V) ∧ ((2nd ↾
(𝑋 × 𝑆)):(𝑋 × 𝑆)⟶𝑆 ∧ ∀𝑥 ∈ 𝑆 (((0g‘𝐺)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦(+g‘𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑦(2nd ↾ (𝑋 × 𝑆))(𝑧(2nd ↾ (𝑋 × 𝑆))𝑥)))))) |
52 | 2, 50, 51 | sylanbrc 695 |
1
⊢ ((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) → (2nd ↾ (𝑋 × 𝑆)) ∈ (𝐺 GrpAct 𝑆)) |